Remark 15 From a logical point of view, the appearance of \(\mathsf {K}:\mathbf {DL}\rightarrow \mathbf {BA}\)  in Thm REF  tells us that, given \(A\in \mathbf {DL}\) , we are only allowed to build a formula \(\Box a\) , \(a\in A\) , if \(a\)  lies in a boolean subalgebra (ie \(a\)  has a complement). This side condition takes us out of the realm of equational logic and, hence, of modal logics given by axioms.
This is related to the fact that \(\mathsf {K}\)  is not strongly finitary [1]} and, therefore, functors involving \(\mathsf {K}\)  cannot be expected to have a presentation by operations and equations.

